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Substitution box (S-Box) has a prominent significance being the fundamental nonlinear component of block cipher which fulfils confusion, one of the properties proposed by Claude Shannon in 1949. In this paper, we proposed an S-Box by using the action of modular group PSL<inline-formula> <tex-math notation="LaTeX">$\left ({2,\mathbb {Z} }\right)$ </tex-math></inline-formula> on projective line PL<inline-formula> <tex-math notation="LaTeX">$\left ({F_{257} }\right)$ </tex-math></inline-formula> over Galois field GF<inline-formula> <tex-math notation="LaTeX">$\left ({2^{8} }\right)$ </tex-math></inline-formula>. In the first step we obtained elements of GF<inline-formula> <tex-math notation="LaTeX">$\left ({2^{8} }\right)$ </tex-math></inline-formula> by using powers of <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> is the primitive root of irreducible polynomial <inline-formula> <tex-math notation="LaTeX">$p\left ({x }\right)$ </tex-math></inline-formula> of order 8 over field <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}_{2}$ </tex-math></inline-formula>, then applied the generators of PSL<inline-formula> <tex-math notation="LaTeX">$\left ({2,\mathbb {Z} }\right)$ </tex-math></inline-formula> and followed steps to get rid of infinity from output. In the final step of proposed scheme, one of the permutations of <inline-formula> <tex-math notation="LaTeX">$S_{16}$ </tex-math></inline-formula> is applied which enhanced the possible number of S-Boxes obtained by any single specific irreducible polynomial <inline-formula> <tex-math notation="LaTeX">$p(x)$ </tex-math></inline-formula> over field <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}_{2}$ </tex-math></inline-formula> of order 8. We analyzed performance of the proposed <inline-formula> <tex-math notation="LaTeX">$8\times 8$ </tex-math></inline-formula> S-Box under cryptographic properties such as strict avalanche criterion, bit independence criterion, nonlinearity, differential approximation probability, linear approximation probability; and compared obtained results with a number of renowned S-Boxes. Lastly, we performed statistical analysis (which comprises of contrast analysis, homogeneity analysis, energy analysis, correlation analysis, entropy analysis and mean of absolute deviation analysis) on our proposed S-Box and obtained results have been compared with adequate number of S-Boxes.

Construction of Nonlinear Component of Block Cipher by Action of Modular Group PSL(2, Z) on Projective Line PL(GF(28))

Construction of Nonlinear Component of Block Cipher by Action of Modular Group PSL(2, Z) on Projective Line PL(GF(2⁸))